Injection and suction of an upper-convected maxwell fluid through a porous-walled tube

The steady-state, similarity solutions of the flow of an upper-convected Maxwell fluid through a tube with a porous wall are constructed by asymptotic and numerical analyses as functions of the direction of flow through the tube, the amount of elasticity in the fluid, as measured by the Deborah number De, and the degree of fluid slip along the tube wall. Fluid slip is assumed to be proportional to the local shear stress and is measured by a slip parameter β that ranges between no-slip (β = 1) and perfect slip (β = 0). The most interesting results are for fluid injection into the tube. For β = 1, the family of flows emanating from the Newtonian limit (De = 0) has a limit point where it turns back to lower values of De. These solutions become asymptotic to De = 0) and develop an O(De) boundary layer near the tube wall with singularly high stresses matched to homogeneous elongational flow in the core. This solution structure persists for all nonzero values of the slip parameter. For β ≠ 1, a family of exact solutions is found with extensional kinematics, but nonzero shear stress convected into the tube through the wall. These flows differ for low De from the Newtonian asymptote only by the absence of the boundary layer at the tube wall. Finite difference calculations evolve smoothly between the Newtonian-like and extensional solutions because of approximation error due to under-resolution of the boundary layer. The radial gradient of the axial normal stress of the extensional flow is infinite at the centerline of the tube for De > 1; this singularity causes failure of the finite difference approximations for these Deborah numbers unless the variables are rescaled to take the asymptotic behavior into account.     

Kinematics and normal stress differences of a viscoelastic fluid undergoing wiggle flow

An experimental program was carried out to determine the laminar regime kinematics and normal stress differences of a viscoelastic fluid in wiggle flow employing non-contact measurement techniques. The viscoelastic fluid was a 5% by weight solution of polyisobutylene dissolved in Primol 355, a high purity mineral oil.The kinematics were determined by Laser-Doppler Anemometry and compared with the data obtained for a Newtonian fluid, Primol 355, under identical flow conditions. It was found that the normalized axial velocity versus axial position curves along the centerline for both fluids superimposed at very low flow rates, an experimental verification that a viscoelastic fluid behaves like a Newtonian fluid under very low shear rates. However, at higher flow rates the behaviour of the viscoelastic fluid curves changed appreciably whereas the Newtonian fluid curves did not change at all. Thus, the effect of flow rate on viscoelastic fluid behaviour was also experimentally established.The normal stress differences were determined using a stress-birefringence apparatus. Data obtained along the centerline clearly exhibited a delayed growth of stress which should be attributed to the expected memory effects in viscoelastic fluid flow.     

The effects of the side walls on the flow of a second grade fluid in ducts with suction and injection

The effects of the side walls on the flow in ducts with suction and injection are examined. Three illustrative examples are given. The first example considers the effect of the side walls on the flow over a porous plate. The second example considers the flow between two parallel porous plates and the third example is devoted to the investigation of the flow in a rectangular duct with two porous walls. Exact solution of the governing equation using the no-slip boundary condition and an additional condition are obtained. The expression of the velocity, the volume flux and the vorticity are given. It is found that for large values of the cross-Reynolds number near the suction region the flow for a Newtonian fluid does not satisfy the boundary condition, but it does not behave in the same way for a second grade fluid. Three examples considered show that there are pronounced effects of the side walls on the flows of a second grade fluid in ducts with suction and injection.     

A pseudospectral approach applicable for time integration of linearized N-S operator that removes pole singularity and physically spurious eigenmodes

This paper addresses a modified singularity removal technique for the eigenvalue or optimal mode problems in pipe flow using a pseudospectral method. The current approach results in the linear stability operator to be devoid of any unstable physically spurious modes, and thus, it provides higher numerical stability during time-based integration. The correctness of the numerical operator is established by calculating the known eigenvalues of pipe Poiseuille flow. Subsequently, the optimal modes are determined with Farrell's approach and compared with the existing literature. The usefulness of this approach is further demonstrated in the time-based numerical integration of the linearized Navier-Stokes operator for the adjoint method–based optimal mode determination. The numerical scheme is implemented with the radial velocity-radial vorticity formulation. Even number of Chebyshev-Lobatto grid points are distributed over the domain r∈[−1,1] omitting the centerline, which also efficiently provides higher resolution near the wall boundary. The boundary conditions are imposed with homogeneous wall boundary conditions, whereas the analytic nature of a proper set of base functions enforces correct centerline conditions. The resulting redundancy introduced in the process is eliminated with the proper usage of parity.     

狭窄血管内偶应力流体流动分析

本文考察了血管狭窄对血液流动的影响,血液以偶应力流体表示,并在求解过程中采用了在管壁上流体质点无相对涡量的边界条件,结果表明,和Young的经典工作相比流动阻抗和壁切应力大于同样程度狭窄下牛顿流体的相应值,偶应力流体对狭窄的敏感性大于牛顿流体;在狭窄发展过程中,偶应力流体的流量要小于牛顿流体的流量,和牛顿流体相比,这些结果更符合生理实际。     

Fibre orientation calculation in injection moulding of reinforced thermoplastics

The orientation of short fibres during the filling with reinforced thermoplastics of a tube and a disk is computed. The flow kinematics is obtained using a finite element method with a moving mesh technique in two dimensions. The orientation is calculated with a decoupled method. Results show the evolution of the orientation of individually tracked fibers, especially in the material front region, for a Newtonian and a shear-thinning matrix behavior, and for different thermal conditions and fiber interactions.     

Unsteady flow of a fourth-grade fluid due to an oscillating plate

The governing non-linear high-order, sixth-order in space and third-order in time, differential equation is constructed for the unsteady flow of an incompressible conducting fourth-grade fluid in a semi-infinite domain. The unsteady flow is induced by a periodically oscillating two-dimensional infinite porous plate with suction/blowing, located in a uniform magnetic field. It is shown that by augmenting additional boundary conditions at infinity based on asymptotic structures and transforming the semi-infinite physical space to a bounded computational domain by means of a coordinate transformation, it is possible to obtain numerical solutions of the non-linear magnetohydrodynamic equation. In particular, due to the unsymmetry of the boundary conditions, in numerical simulations non-central difference schemes are constructed and employed to approximate the emerging higher-order spatial derivatives. Effects of material parameters, uniform suction or blowing past the porous plate, exerted magnetic field and oscillation frequency of the plate on the time-dependent flow, especially on the boundary layer structure near the plate, are numerically analysed and discussed. The flow behaviour of the fourth-grade non-Newtonian fluid is also compared with those of the Newtonian fluid.     

低雷诺数下附面层组合抽吸方案对压气机特性影响的研究

为了分析附面层抽吸流动控制对低雷诺数下压气机特性的影响,本文采用数值方法模拟了低雷诺数下附面层组合抽吸方案对NASA Rotor 37跨音速压气机性能和稳定性的影响特点及作用机理。通过在该压气机转子叶片吸力面和机匣上分别设计附面层抽吸槽,探讨了组合抽吸方案对低雷诺数下(H=20km)压气机性能和稳定性的影响。结果表明:采用组合抽吸方案后,压气机峰值效率提高约1.3%;压气机最大增压比提高约2.5%;压气机转子的近失速点流量减小约14.6%。进一步分析作用机理发现,组合抽吸槽有效抑制了附面层径向涡向叶顶的运动和聚集,使叶顶附面层分离区减少约70%从而有效改善了压气机的流场特性。     

Injection/suction boundary conditions for fluid–structure interaction simulations in incompressible flow

This paper presents the analysis of injection/suction boundary conditions in the context of the fluid–structure interactions simulation of the incompressible turbulent flow. First, the equations used in the modelling of the fluid and the structure are presented, as well as the numerical methods used in the corresponding solvers. Injection/suction boundary conditions are then presented with details of different implementation alternatives. Arbitrary Lagrangian–Eulerian (ALE) approach was also implemented in order to test the injection/suction boundary conditions. Numerical tests are performed where injection/suction boundary conditions are compared to ALE simulations. These tests include forced movement of the structure and two‐degrees‐of‐freedom structure model simulations. Copyright © 2002 John Wiley & Sons, Ltd.     

Effects of thermal stratification on flow and heat transfer past a porous vertical stretching surface

The aim of this paper is to present the boundary layer flow of viscous incompressible fluid due to a porous vertical stretching surface with a power-law stretching velocity in a thermally stratified medium. Using a special form of Lie group transformations viz. scaling group of transformations, similarity solutions for this problem are obtained. The equations are then solved numerically. With increasing values of the stratification parameter, the velocity as well as temperature decreases. At a particular point of the porous stretching sheet, the velocity decreases with the increasing suction parameter. The dimensionless temperature at a point of the sheet decreases due to suction but increases due to injection. The findings of this study reveal that stratification and suction can be used as means of cooling the boundary layer flow region.     

Selective withdrawal of polymer solutions: Experiments

Selective withdrawal refers to the process of drawing one or both components of stratified fluids through a tube placed near their interface. This paper reports an experimental study of selective withdrawal of viscous and viscoelastic liquids under air. The key mechanism of interest is how the viscoelasticity in the bulk liquid affects the evolution of the free surface. This is investigated by comparing the interfacial behavior between a Newtonian silicone oil and two dilute polymer solutions. While the surface undergoes smooth and gradual deformation for Newtonian liquids, for the polymer solutions there is a critical transition where the surface forms a cusp from which an air jet emanates toward the suction tube. This transition shows a hysteresis when the flow rate or location of the tube is varied. In the subcritical state, the surface of polymer solutions deform much more than its Newtonian counterpart but the deformation is more localized. The interfacial behavior of the polymer solutions can be attributed to the large polymer stress that develops under the surface because of predominantly extensional deformation.     

Newtonian flow in a triangular duct with slip at the wall

We consider the Newtonian Poiseuille flow in a tube whose cross-section is an equilateral triangle. It is assumed that boundary slip occurs only above a critical value of the wall shear stress, namely the slip yield stress. It turns out that there are three flow regimes defined by two critical values of the pressure gradient. Below the first critical value, the fluid sticks everywhere and the classical no-slip solution is recovered. In an intermediate regime the fluid slips only around the middle of each boundary side and the flow problem is not amenable to analytical solution. Above the second critical pressure gradient non-uniform slip occurs everywhere at the wall. An analytical solution is derived for this case and the results are discussed.     

Analysis of a laminar boundary layer flow over a flat plate with injection or suction

An analysis is performed to study a laminar boundary layer flow over a porous flat plate with injection or suction imposed at the wall. The basic equations of this problem are reduced to a system of nonlinear ordinary differential equations by means of appropriate transformations. These equations are solved analytically by the optimal homotopy asymptotic method (OHAM), and the solutions are compared with the numerical solution (NS). The effect of uniform suction/injection on the heat transfer and velocity profile is discussed. A constant surface temperature in thermal boundary conditions is used for the horizontal flat plate.     

刚性圆管中血液周期振荡流的切应力分布

本文通过求解圆管内血液振荡流的基本方程，求得圆管内血液流的压力梯度与切应力之间的关系式。在此基础上，详细讲座了圆管中轴向流速和切变率谐波的变化规律，指出流速谐波和切变率谐波的幅值都将随着谐波次数的增大而逐渐减小。为了使所得结果便于应用。文章通过管轴向中心线流速与压力梯度之间的关系式，进一步给出一种利用管轴向中心线流速计算管内切应力分布的简便方法。该方法用于检测活体血管内血液振荡流的切应力分布，具有操作简单，精度较高的优点。最后，以人体颈动脉为例，讨论血液周期振荡流的切应力的分布特性。发现在任意时刻，除了邻近管壁处切应力急剧增大到一定数值之外，沿管截面切应力分布相当均匀且接近于零，呈现出与定常流不同的切应力分布特征。     

Anisotropy measurements in the boundary layer over a flat plate with suction

Laser Doppler velocity measurements are carried out in a turbulent boundary layer subjected to concentrated wall suction (through a porous strip). The measurements are taken over a longitudinal distance of 9× the incoming boundary layer thickness ahead of the suction strip. The mean and rms velocity profiles are affected substantially by suction. Two-point measurements show that the streamwise and wall-normal autocorrelations of the streamwise velocity are reduced by suction. It is found that suction alters the redistribution of the turbulent kinetic energy k between its components. Relative to the no-suction case, the longitudinal Reynolds stress contributes more to k than the other two normal Reynolds stresses; in the outer region, its contribution is reduced which suggests structural changes in the boundary layer. This is observed in the anisotropy of the Reynolds stresses, which depart from the non-disturbed boundary layer. With suction, the anisotropy level in the near-wall region appears to be stronger than that of the undisturbed layer. It is argued that the mean shear induced by suction on the flow is responsible for the alteration of the anisotropy. The variation of the anisotropy of the layer will make the development of a turbulence model quite difficult for the flow behind suction. In that respect, a turbulence model will need to reproduce well the effects of suction on the boundary layer, if the model is to capture the effect of suction on the anisotropy of the Reynolds stresses.     

Laminar flow in the inlet region of a porous tube

Integral forms of the boundary layer equations, coupled with an assumed n-th degree boundary layer velocity profile, are used to study the effect of injection on the laminar flow in the inlet region of a circular tube. Results are illustrated graphically for a fourth degree velocity profile. It is found that the length of the inlet region decreases with the injection parameter and also that, at a given distance from the entry, the pressure drop in the inlet region increases with the injection parameter.     

MODELLING AND CALCULATION OF THE TURBULENT BOUNDARY LAYER ON A ROTATING CYLINDER SURFACE WITH STRONG SUCTION 1)

提出了湍流边界层的一种简单、快速计算方法, 用以求解强吸气作用下旋转圆筒表面边界层流动. 首先, 理论分析了同心圆筒间的旋转流体运动, 外筒静止、内筒旋转且为多孔吸气条件. 强吸气情况下旋转流动主要表现为内筒壁面附近的边界层流动, 基于这一事实得到了周向速度分布的解析表达式. 其次, 通过引入新参数扩展Cebeci-Smith代数湍流模型, 使其能考虑流线曲率、壁面吸气、低Reynolds数效应等因素. 针对这些因素的综合影响, 采用解析修正和经验参数对模型进行调整. 同时, 基于Reynolds应力湍流模型的仿真结果, 校准代数湍流模型中的经验参数. 最后, 给出基于广义Cebeci-Smith湍流模型的旋转壁面边界层流动的迭代算法, 该算法适用于需要特殊迭代过程的轴向及周向流动均匀情况. 计算了不同旋转速度和吸气强度组合工况下的边界层流动, 其周向速度和湍流强度分布与基于Reynolds应力湍流模型的计算结果非常接近. 并且表明, 当Reynolds应力湍流模型数值模拟预测内筒边界层为稳定层流时, 该方法也再现了相同初始条件下的层流边界层.     

Near-field measurements and development of a new boundary layer over a flat plate with localized suction

Suction on a turbulent boundary layer is applied through a narrow strip in order to understand the effects suction can have on the boundary layer development and turbulent structures in the flow. Detailed two-component laser Doppler velocimetry (LDV) and laser-induced fluorescence (LIF) based measurements have been undertaken in regions close to the suction strip and further downstream. The region close to the strip involves a flow reversal accompanied by a change in sign for the Reynolds shear stress and strong gradients in the flow variables. The mean streamwise velocity after suction remains larger than its corresponding no-suction value. Relative to the no-suction case, the velocity fluctuations first decrease with suction followed by a slow recovery which may involve a slight overshoot. LIF visualizations indicate that compared to the no-suction case, the low-speeds streaks stay closer to the wall and exhibit a smaller amount of spanwise and wall-normal oscillations with suction. The visualization results are consistent with two-point velocity correlation measurements. The streamwise and spanwise correlation measurements indicate that the structures are disrupted or removed from the boundary layer due to suction suggesting that the original boundary layer has been strongly influenced by suction. The results are explained by the development of a new inner layer that forms downstream of the suction strip.     

强吸气旋转圆筒壁面湍流边界层建模及计算

提出了湍流边界层的一种简单、快速计算方法, 用以求解强吸气作用下旋转圆筒表面边界层流动. 首先, 理论分析了同心圆筒间的旋转流体运动, 外筒静止、内筒旋转且为多孔吸气条件. 强吸气情况下旋转流动主要表现为内筒壁面附近的边界层流动, 基于这一事实得到了周向速度分布的解析表达式. 其次, 通过引入新参数扩展Cebeci-Smith代数湍流模型, 使其能考虑流线曲率、壁面吸气、低Reynolds数效应等因素. 针对这些因素的综合影响, 采用解析修正和经验参数对模型进行调整. 同时, 基于Reynolds应力湍流模型的仿真结果, 校准代数湍流模型中的经验参数. 最后, 给出基于广义Cebeci-Smith湍流模型的旋转壁面边界层流动的迭代算法, 该算法适用于需要特殊迭代过程的轴向及周向流动均匀情况. 计算了不同旋转速度和吸气强度组合工况下的边界层流动, 其周向速度和湍流强度分布与基于Reynolds应力湍流模型的计算结果非常接近. 并且表明, 当Reynolds应力湍流模型数值模拟预测内筒边界层为稳定层流时, 该方法也再现了相同初始条件下的层流边界层.